The latest version of Google Earth has ocean topography. I spent about 20 minutes, and I found a point at -33,038 feet. Can anyone beat that? Now, I’ve got to dig up my copy of Tales of Topographic Oceans.
Just once? In 1960? 
Something told me that the Mariana Trench is what I should be looking for (I believe it’s been mentioned on these boards once or twice). I found the trench, but it’s not exactly a small trench. Finding the deepest point is like finding a needle in a haystack. I spent about 10 minutes searching around (I don’t have the willpower that you do), and I couldn’t find anything deeper than about 9,000 metres (30,000 feet), so I gave up and looked up the coordinates of the deepest point in Wikipedia.
11°19′N 142°15′E
Even armed with these coordinates, I didn’t find anything deeper than 10,342 metres (33,930 feet), but Wikipedia says it should be 10,900 metres (35,760 ft).
If you zoom in close enough to the coordinates in the spoiler, a Wikipedia icon pops up for the Bathyscaphe Trieste, which performed the infamous descent. Nearby is a Wikipedia icon for Challenger Deep. Poking around between these two icons I managed to find a point at -10,863 metres (-35,640 feet).
I have found -10.889 meter at 1121’30,50’'N and 14233’20,83’'E
I have found -10.889 meter at 1121’30,50’'N and 14233’20,83’'E
I have found -35.726 feet at 1121’30,50’'N and 14233’20,83’'E
Try this spot (converted): 11°21’30.50N, 142°33’20.83E